7 Enhanced regularization

Regularization is key to subsurface imaging

\(\Rightarrow\) Add any information about the subsurface:

correlation lengths and angles
structural boundaries (from boreholes, seismics, GPR)
point information (samples, borehole logs, )
limits of the possible parameters (e.g. positivity, natural bounds)

7.1 Smoothness constraints

4 model cells, 3 boundaries

1st order derivative \[ \vb C = \mqty(-1 & 1 & 0 & 0\\0 & -1 & 1 & 0\\0 & -1 & 0 & 1) \] can be weighted by

inverse midpoint distance
boundary size (length, area)
information about boundaries

7.2 Anisotropic smoothness

7.3 Structural decoupling

Structurally constrained inversion (Bergmann *et al.* 2013)

Derivation of smoothness matrix (Wagner and Uhlemann 2021)

Seismic cross-section (Tanner *et al.* 2020)

ERT inversion result without (a) and with (b) structural constraints from seismic (Tanner *et al.* 2020)

7.4 Selected smoothness

7.5 Model reduction

7.6 Region-wise settings

Placement of mesh divided into different regions in greater for

Mesh with different regions (Coscia *et al.* 2011)

Cross-section with decoupled (a) and coupled (b) regions (Coscia *et al.* 2011)

7.7 Geostatistical regularization

\(\vb M_{ij}=\sigma^2\exp(-3\sqrt{\frac{|x_i-x_j|}{I_x}^2+\frac{|y_i-y_j|}{I_y}^2})\quad\Rightarrow\quad \vb C=\vb M^{-0.5}\)

Geostatistical correlation function (left) and regularization operator (right) after Jordi *et al.* (2018)

7.8 Geostatistical regularization - example

Synthetic model (a) and inversion results with classic smoothness constraints (b), true correlation (c) and slightly wrong correlation (d) (Jordi *et al.* 2018)

Bergmann P, Ivandic M, Norden B, Rücker C, Kiessling D, Lüth S, Schmidt-Hattenberger C and Juhlin C. 2013. Combination of seismic reflection and constrained resistivity inversion with an application to 4D imaging of the CO2 storage site, ketzin, germany. Geophysics. 79. B37–50

Coscia I, Greenhalgh S, Linde N, Doetsch J, Marescot L, Günther T and Green A. 2011. 3D crosshole apparent resistivity static inversion and monitoring of a coupled river-aquifer system. Geophysics. 76. G49–59

Jordi C, Doetsch J, Günther T, Schmelzbach C and Robertsson J O A. 2018. Geostatistical regularization operators for geophysical inverse problems on irregular meshes. Geophysical Journal International. 213. 1374–86

Tanner D C, Buness H, Igel J, Günther T, Gabriel G, Skiba P, Plenefisch T, Gestermann N and Walter T R. 2020. Chapter 3 - fault detection. Understanding faults ed D Tanner and C Brandes (Elsevier) pp 81–146 Online: http://www.sciencedirect.com/science/article/pii/B9780128159859000035

Wagner F M and Uhlemann S. 2021. Advances in geophysics. Inversion of geophysical data (Elsevier) pp 1–72